The average revenues of a company over 13 consecutive years is Rs 78 lakhs. The average revenue of the first 7 years is Rs 73 lakhs and the average revenue of the last 7 years is Rs 85 lakhs. What is the revenue (in lakhs) in the 7th year?

Introduction / Context:
This question is similar to other overlapping average problems. We are given the overall average for 13 consecutive years and separate averages for the first 7 years and the last 7 years. Since the 7th year appears in both the first and last group, we can use this overlap to find the revenue in that specific year.

Given Data / Assumptions:

• Number of years = 13.
• Average revenue for all 13 years = Rs 78 lakhs.
• Average revenue for first 7 years = Rs 73 lakhs.
• Average revenue for last 7 years = Rs 85 lakhs.
• The 7th year is common to both sets of 7 years.
• We are required to find the revenue in the 7th year.

Concept / Approach:
Let the revenues be R1, R2, ..., R13. The overall average gives the total of all 13 years. The average of first 7 years gives R1 + ... + R7. The average of last 7 years gives R7 + ... + R13. Adding these two sums counts all 13 years once and the 7th year twice. We can therefore write an equation where the sum of first 7 years plus the sum of last 7 years equals the total for 13 years plus the 7th year revenue. Solving that equation gives us R7.

Step-by-Step Solution:
Total revenue for 13 years = 78 * 13 = Rs 1014 lakhs. Sum of first 7 years = 73 * 7 = Rs 511 lakhs. Sum of last 7 years = 85 * 7 = Rs 595 lakhs. Add these sums: 511 + 595 = Rs 1106 lakhs. This combined sum includes the revenues of all 13 years once, plus the 7th year one extra time. So, 1106 = 1014 + R7. R7 = 1106 - 1014 = Rs 92 lakhs.
Verification / Alternative check:
If R7 = Rs 92 lakhs, then sum of first 7 years is 511, so R1 + R2 + R3 + R4 + R5 + R6 = 511 - 92 = Rs 419 lakhs. Sum of last 7 years is 595, so R8 + R9 + R10 + R11 + R12 + R13 = 595 - 92 = Rs 503 lakhs. Total for all 13 years = 419 + 92 + 503 = Rs 1014 lakhs. Average = 1014 / 13 = 78 lakhs, which matches the overall average given.
Why Other Options Are Wrong:
Option A (Rs 94 lakhs): Makes the combined sums inconsistent with the overall total of 1014 lakhs. Option B (Rs 90 lakhs): Too low to satisfy both subset averages at the same time. Option C (Rs 88 lakhs): Also fails to keep both partial averages correct. Option D (Rs 92 lakhs): The only value consistent with all the given averages, so it is correct.
Common Pitfalls:
A common error is to simply take the average of 73 and 85 and treat that as the 7th year revenue, which is not valid. Some students forget that the combined sum of the first 7 and last 7 years counts the 7th year twice. Mistakes in basic multiplication or subtraction can also lead to incorrect values for R7.
Final Answer:
The revenue in the 7th year is Rs 92 lakhs.